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Mirrors > Home > MPE Home > Th. List > Mathboxes > bj-1uplex | Structured version Visualization version GIF version |
Description: A monuple is a set if and only if its coordinates are sets. (Contributed by BJ, 6-Apr-2019.) |
Ref | Expression |
---|---|
bj-1uplex | ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | bj-pr11val 34441 | . . 3 ⊢ pr1 ⦅𝐴⦆ = 𝐴 | |
2 | bj-pr1ex 34442 | . . 3 ⊢ (⦅𝐴⦆ ∈ V → pr1 ⦅𝐴⦆ ∈ V) | |
3 | 1, 2 | eqeltrrid 2895 | . 2 ⊢ (⦅𝐴⦆ ∈ V → 𝐴 ∈ V) |
4 | df-bj-1upl 34434 | . . 3 ⊢ ⦅𝐴⦆ = ({∅} × tag 𝐴) | |
5 | p0ex 5250 | . . . 4 ⊢ {∅} ∈ V | |
6 | bj-xtagex 34425 | . . . 4 ⊢ ({∅} ∈ V → (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V)) | |
7 | 5, 6 | ax-mp 5 | . . 3 ⊢ (𝐴 ∈ V → ({∅} × tag 𝐴) ∈ V) |
8 | 4, 7 | eqeltrid 2894 | . 2 ⊢ (𝐴 ∈ V → ⦅𝐴⦆ ∈ V) |
9 | 3, 8 | impbii 212 | 1 ⊢ (⦅𝐴⦆ ∈ V ↔ 𝐴 ∈ V) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 209 ∈ wcel 2111 Vcvv 3441 ∅c0 4243 {csn 4525 × cxp 5517 tag bj-ctag 34410 ⦅bj-c1upl 34433 pr1 bj-cpr1 34436 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1797 ax-4 1811 ax-5 1911 ax-6 1970 ax-7 2015 ax-8 2113 ax-9 2121 ax-10 2142 ax-11 2158 ax-12 2175 ax-ext 2770 ax-rep 5154 ax-sep 5167 ax-nul 5174 ax-pow 5231 ax-pr 5295 ax-un 7441 |
This theorem depends on definitions: df-bi 210 df-an 400 df-or 845 df-3an 1086 df-tru 1541 df-fal 1551 df-ex 1782 df-nf 1786 df-sb 2070 df-mo 2598 df-eu 2629 df-clab 2777 df-cleq 2791 df-clel 2870 df-nfc 2938 df-ne 2988 df-ral 3111 df-rex 3112 df-rab 3115 df-v 3443 df-sbc 3721 df-csb 3829 df-dif 3884 df-un 3886 df-in 3888 df-ss 3898 df-nul 4244 df-if 4426 df-pw 4499 df-sn 4526 df-pr 4528 df-op 4532 df-uni 4801 df-br 5031 df-opab 5093 df-xp 5525 df-rel 5526 df-cnv 5527 df-dm 5529 df-rn 5530 df-res 5531 df-ima 5532 df-bj-sngl 34402 df-bj-tag 34411 df-bj-proj 34427 df-bj-1upl 34434 df-bj-pr1 34437 |
This theorem is referenced by: bj-2uplex 34458 |
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